Mathematical Sciences: The Geometry and Topology of Instantons and Related Problems

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The principal investigator will expand upon his recent discovery of new homotopy-theoretic structures on complex surfaces. This interdisciplinary project involves the use of algebraic topology to understand the structure of the solution space of certain systems of nonlinear partial differential equations which appear naturally in differential geometry and mathematical physics. Additionally, he will investigate several problems involving iterated loop space structures. Two shapes are homotopic if closed loops on them have identical collapsing properties. For example, a circle and the shape of a solid coffee mug (with one handle) are homotopy equivalent. But if the mug were to have two handles or no handles, it would not be homotopy equivalent to a circle. This concept has been explored by mathematicians for over a hundred years, but its use in problems related to mathematical physics has been a recent phenomena. The principal investigator will contribute to this growing interdisciplinary activity.